RealRooted

RealRooted is an experimental Lean 4 library for real-rooted univariate polynomials, interlacing, compatibility, Polya-frequency sequences, and related combinatorial applications.

The repository is a research formalization workspace rather than a polished mathlib contribution. The useful part is that the named theorem declarations below are checked by Lean, and the surrounding files give searchable proof infrastructure for real-rootedness and interlacing arguments. The long-term goal is to extract stable, reusable components for eventual upstreaming.

Table of Contents

• Build
• Repository Layout
• Main Concepts
• Checked Highlights
• Current Roadmap
• Development Notes
• Bibliography and Links

Build

The project uses Lean 4 and Mathlib through Lake.

lake exe cache get
lake build

Useful focused checks for recent theorem areas are:

lake build RealRooted.CommonInterleaverTwo
lake build RealRooted.ChudnovskySeymour
lake build RealRooted.Hadamard
lake build RealRooted.VeroneseMatrix
lake build RealRooted.VeroneseSection
lake build RealRooted.Bezoutian

Repository Layout

• RealRooted.lean is the umbrella import for the public development.
• RealRooted/Basic.lean, Derivative.lean, Wagner*.lean, and InterlacingSequence*.lean contain the core interlacing API.
• RealRooted/CommonInterleaver*.lean and ChudnovskySeymour.lean contain the compatibility and common-interleaver work.
• RealRooted/AissenSchoenbergWhitney.lean, PFPolynomial.lean, VeroneseSection.lean, and VeroneseMatrix.lean contain the PF, Toeplitz, and Veronese-section material.
• RealRooted/SymmetricDecomposition.lean, Bezoutian.lean, and Hadamard.lean contain larger theorem packages and classical interfaces.
• RealRooted/Challenges/ contains compact entry points for famous theorem statements, each linking the Lean-facing declaration to human catalog statements and references.
• RealRooted/CombinatorialExamples/ contains examples such as Eulerian, type B Eulerian, simsun, Touchard, Narayana, Motzkin, and related families.
• RealRooted/Mathlib/ contains local compatibility lemmas intended to look like future Mathlib additions.

Main Concepts

• Interlaces f g, Prec f g, and Prec0 f g: the main interlacing and proper-position relations. Prec0 is the zero-aware version.
• p = 0 ∨ p.Splits: the zero-aware real-rootedness convention used in closure statements where the zero polynomial is a natural exceptional case.
• IsGeneralizedSturmSeq ps, IsInterlacingSeq fs, and IsInterlacingSeq0 fs: list-level Sturm and interlacing predicates.
• Compatible f g, PairwiseCompatible fs, and FamilyCompatible fs: Chudnovsky-Seymour style compatibility predicates.
• HasCommonInterleaver fs and HasCommonLeftInterleaver fs: common interleaver data for finite families.
• AllComboRealRooted f g: every real linear combination of f and g is zero or real-rooted.
• IsPolyaFreqSeq a: total nonnegativity of the Toeplitz matrix of a sequence.
• veroneseSectionPolynomial r k p: the fixed-residue Veronese section of a polynomial.
• FullyInterlacingPair a b: the two-row Lace total-nonnegativity interface used by the Veronese and Hurwitz-matrix route.

Checked Highlights

Every declaration named in this section is a checked Lean declaration, unless it is explicitly described as an unproved conjecture (bearing a sorry stub).

Theorem-Level Highlights

The formalization now contains theorem interfaces for several standard real-rootedness and interlacing results. In ordinary mathematical language, the checked or challenge-facing highlights are:

• Cauchy interlacing: the eigenvalues of a principal codimension-one submatrix of a real symmetric or Hermitian matrix interlace the eigenvalues of the original matrix. See cauchy_interlacing and RealRooted.Challenges.CauchyInterlacing; references include Fisk (2005) and Godsil (2017).
• Wagner's lemma: for real-rooted polynomials with nonpositive roots and positive leading coefficients, common interlacers are closed under addition, and multiplication by X shifts the interlacing relation in the expected way. See RealRooted.Challenges.Wagner; reference: Wagner (1992).
• Obreschkoff's theorem: two polynomials have a real-rooted real pencil alpha * f + beta * g if and only if they interlace, up to orientation and degree conventions. See allComboRealRooted_of_prec, prec_of_allComboRealRooted, and RealRooted.Challenges.Obreschkoff; references include Obreschkoff (1963), Dedieu (1992), and Branden (2004).
• Interlacing preservers: a linear operator that preserves real-rootedness sends interlacing pairs to interlacing pairs, up to the zero and orientation conventions used by Prec0. See operatorPreservesInterlacingPairsUpToOrder and RealRooted.Challenges.OperatorPreservers; reference: Branden (2004).
• Matrix preservers: a polynomial matrix with nonnegative coefficients preserves interlacing sequences when its two-by-two affine tests interlace. See matrix_preserves_interlacing_seq and RealRooted.Challenges.MatrixInterlacing; reference: Branden (2015).
• Favard interlacing: a three-term Favard recurrence with positive recurrence coefficients gives a Sturm sequence, hence every polynomial in the sequence is real-rooted. See favardInterlacing and RealRooted.Challenges.Favard; reference: Favard (1935).
• Polya-frequency and Veronese results: the ASW reverse direction and the Toeplitz/PF infrastructure imply that Veronese sections of a real-rooted polynomial with nonnegative coefficients are real-rooted or zero. See aissenSchoenbergWhitney_reverse, isRealRootedOrZero_veroneseSectionPolynomial_of_realRooted_nonneg_matrix, and RealRooted.Challenges.VeroneseSections; references include Aissen--Schoenberg--Whitney (1952) and Athanasiadis--Wagner (2024).
• Eulerian examples: ordinary Eulerian and type B Eulerian polynomials are real-rooted, and the formalization proves stronger consecutive interlacing or Sturm-sequence statements. See RealRooted.Challenges.Eulerian; references include Frobenius (1910).
• Graph polynomials: finite claw-free graphs have real-rooted independence polynomials, and matching-polynomial corollaries are packaged through the line-graph reduction. See Graph.clawFree_indepPoly_splits and RealRooted.Challenges.ChudnovskySeymour; references include Chudnovsky--Seymour (2007) and Heilmann--Lieb (1972).

The challenge surface also records open theorem-shaped targets. Kurtz's coefficient inequality criterion and the Hermite--Poulain differential-operator preserver are stated as sorry targets, while Borcea--Branden's finite-symbol classification is currently a scaffold because a faithful statement needs a multivariate real-stability API.

Interlacing And Preservers

• derivative_interlaces: Rolle-style derivative interlacing for real-rooted polynomials.
• prec_add_of_prec_right_of_posLeadingCoeff, prec_add_of_prec_left, and prec0_mul_X_of_prec0: checked Wagner-lemma forms for common interlacers and multiplication by X.
• prec_ma_wang and generalizedLiuWangCriterion: Ma-Wang and Liu-Wang style criteria for interlacing recurrences and weighted sums.
• favardInterlacing and isRealRooted_of_favard: a Favard recurrence interface for orthogonal-polynomial style Sturm sequences.
• matrix_preserves_interlacing_seq and matrix_preserves_interlacing_seq0_of_2x2: matrix preservers from finite two-by-two interlacing checks.
• operatorPreservesInterlacingPairsUpToOrder: a general operator-preserver interface for interlacing pairs.
• cauchy_interlacing: Cauchy's eigenvalue interlacing theorem for Hermitian matrices and one-index principal submatrices.

Compatibility And Common Interleavers

• hasCommonInterleaver_of_pairwiseHasCommonInterleaver: pairwise common interleavers imply a global common interleaver.
• isRealRooted_sum_of_commonInterleaver and isRealRooted_sum_of_commonLeftInterleaver: nonnegative sums are real-rooted when a family has common interleaver data.
• familyCompatible_of_commonInterleaver and pairwiseCompatible_of_familyCompatible: the easy directions relating common interleavers and compatibility.
• Declarations with prefix chudnovskySeymour_fourWay_of_: several checked Chudnovsky-Seymour four-way packages under formalized bridge hypotheses.
• pairwiseCompatible_iff_familyCompatible_of_natDegree_le_one: the complete low-degree compatibility equivalence.

Symmetric Decomposition And Bezoutians

• idDecompositionExistsUnique and rdDecompositionExistsUnique: existence and uniqueness of the Branden-Solus symmetric decompositions.
• brandenSolusTheorem26: the formalized Branden-Solus Theorem 2.6 package in the local Prec language.
• isRealRooted_fPolynomial_of_isRealRooted_of_hasNonnegCoeffs: real-rootedness preservation for the h -> f transform in the nonnegative setting.
• strictPrecSameDegree_iff_bezoutMatrix_posDef: strict same-degree Bezoutian characterization.

Polya Frequency, ASW, And Veronese Sections

• aissenSchoenbergWhitney_reverse: the reverse Aissen-Schoenberg-Whitney direction, from real-rooted nonpositive roots and nonnegative coefficients to a Polya-frequency coefficient sequence.
• aissenSchoenbergWhitneyForward: the target theorem for the opposite ASW direction.
• IsPolyaFreqSeq.veroneseSectionSeq and IsPolyaFreqSeq_veroneseSectionPolynomial_coeff: Veronese subsequences and Veronese section coefficients preserve Toeplitz total nonnegativity.
• isRealRootedOrZero_veroneseSectionPolynomial_of_realRooted_nonneg_matrix: the completed cyclic-matrix proof that Veronese sections of a real-rooted nonnegative-coefficient polynomial are zero or real-rooted.
• not_isUpperHalfPlaneStable_hermiteBiehlerPolynomial_X_neg_one: a checked counterexample documenting why the Hermite-Biehler forward route is exposed only in sign-normalized form.

Combinatorial Examples

The example files prove real-rootedness and, in many cases, Sturm or interlacing sequence statements for standard combinatorial families. Representative declarations include:

• isRealRooted_eulerianTilde
• isRealRooted_typeBEulerian
• isRealRooted_simsun
• isRealRooted_touchard
• isRealRooted_coloredSetPartitions
• isRealRooted_narayana_of_nonnegCoeffs
• isRealRooted_motzkin

Current Roadmap

The Chudnovsky-Seymour and Heilmann-Lieb graph-facing line is now represented by checked theorem interfaces in ChudnovskySeymour.lean and HeilmannLieb.lean. The graph endpoint is Graph.clawFree_indepPoly_splits: finite claw-free graphs have real-rooted independence polynomials. The matching-polynomial corollaries are packaged through the line-graph reduction in HeilmannLieb.

The next standard theorem input is Garloff-Wagner Hadamard proper-position, recorded as garloffWagnerHadamardNonnegPrec. This is the remaining RealRooted theorem currently used as an external standard fact by the SuperEulerian project.

New formalization target: Braun-Jal, Order polytopes of generalized snake posets are h^-real-rooted*, arXiv:2607.00922v1. The immediate Lean todo is their Theorem 4.1: for a generalized snake word w of length n >= 1, the non-nesting rook polynomial M_{epsilon w} is real-rooted and the polynomial for the word with the final letter deleted interlaces it. Concrete subtasks: define generalized snake words, deletion of the final letter, and prefix operations; define the associated squarecase/non-nesting rook polynomials; formalize the modified Narayana polynomials P_n, the auxiliary sums G_n, and the interlacing inputs behind Lemmas 3.3 and 3.4; prove the positive non-nesting recurrence of Theorem 3.5; expose the Branden matrix interlacing preserver used in the induction; prove Theorem 4.1 by strong induction; then package the resulting h^*-real-rootedness statement for order polytopes via the Stanley / Alexandersson-Jal width-two correspondence.

Short-term documentation/onboarding now uses concise challenge entry-point files in RealRooted/Challenges/ for the famous general theorems and theorem-shaped targets. These files point to the corresponding human theorem statements on symmetricfunctions.com and to the original publications or catalog references, while the detailed proof infrastructure remains in the main theorem modules. The current challenge surface includes ASW, Chudnovsky-Seymour, Hadamard, Wagner, Cauchy interlacing, Obreschkoff, operator and matrix interlacing preservers, Hermite-Biehler/Hurwitz, Veronese sections, Favard, Kurtz, Hermite-Poulain, Borcea-Branden, and Eulerian polynomials.

Longer term, a Borcea-Branden direction would be a substantial expansion toward stability theory. A realistic path would first build the Hermite-Biehler and Hurwitz-matrix total-nonnegativity interfaces into proved theorems, then add the multivariate stability infrastructure needed for algebraic-symbol preserver theorems.

GitHub issues track individual proof tasks rather than being duplicated here. Current open themes include Liu's compatible-sequences theorem, the Gustafsson-Solus interlacing recursion, the Haglund-Zhang s-inversion backend, characteristic-polynomial packaging for Cauchy interlacing, Kurtz and Hermite-Poulain preservers, a finite Borcea-Branden symbol interface, and the Braun-Jal generalized snake poset target.

Development Notes

For unproved conjectures and target theorem interfaces, the project prefers declaring standard Lean theorem signatures with sorry proofs. We avoid passing conjectures around as hypothetical parameters (axioms in disguise). Using standard sorry stubs simplifies downstream signatures, avoids parameter propagation clutter, and aligns with Mathlib best practices.

New Lean code should follow the Lean community style guidelines and Mathlib naming conventions where practical. In particular, keep declarations explicit, prefer small reusable lemmas, keep top-level declarations flush-left, and make sure public modules are imported by RealRooted.lean. All committed code must build without warnings (with the exception of sorry warnings).

Please keep repository configuration files (like lakefile.toml and lake-manifest.json) free of hardcoded absolute paths such as /lake-cache/projects/.... Reusable relative repository paths (e.g. .lake/packages and .lake/build) ensure that the builds work out-of-the-box in local developer environments.

Bibliography and Links

• M. Aissen, I. J. Schoenberg, and A. M. Whitney, On the generating functions of totally positive sequences. I, J. Analyse Math. 2 (1952), 93--103.
• C. A. Athanasiadis and C. H. Wagner, Veronese sections and interlacing matrices of polynomials and formal power series, arXiv:2404.12989.
• J. Borcea and P. Branden, The Lee-Yang and Polya-Schur programs. I. Linear operators preserving stability, Invent. Math. 177 (2009), 541--569.
• P. Branden, On operators on polynomials preserving real-rootedness and the Neggers-Stanley conjecture, J. Algebraic Combin. 20 (2004), 119--130.
• P. Branden, Iterated sequences and the geometry of zeros, J. Reine Angew. Math. 658 (2011), 115--131.
• P. Branden, Unimodality, log-concavity, real-rootedness and beyond, in Handbook of Enumerative Combinatorics, CRC Press, 2015, 437--483.
• B. Braun and A. Jal, Order polytopes of generalized snake posets are h^-real-rooted*, arXiv:2607.00922.
• P. Branden and L. Solus, Symmetric decompositions and real-rootedness, Int. Math. Res. Not. (2019), doi:10.1093/imrn/rnz059.
• M. Chudnovsky and P. Seymour, The roots of the independence polynomial of a clawfree graph, J. Combin. Theory Ser. B 97 (2007), 350--357.
• J.-P. Dedieu, Obreschkoff's theorem revisited: what convex sets are contained in the set of hyperbolic polynomials?, J. Pure Appl. Algebra 81 (1992), 269--278.
• J. Favard, Sur les polynomes de Tchebicheff, C. R. Acad. Sci. Paris 200 (1935), 2052--2053.
• S. Fisk, A very short proof of Cauchy's interlace theorem for eigenvalues of Hermitian matrices, Amer. Math. Monthly 112 (2005), 118.
• F. G. Frobenius, Uber die Bernoullischen Zahlen und die Eulerschen Polynome, Sitzungsberichte der Koniglich Preussischen Akademie der Wissenschaften (1910), 809--847.
• J. Garloff and D. G. Wagner, Hadamard Products of Stable Polynomials Are Stable, J. Math. Anal. Appl. 202 (1996), 797--809.
• C. D. Godsil, Algebraic Combinatorics, Routledge, 2017.
• O. J. Heilmann and E. H. Lieb, Theory of monomer-dimer systems, Comm. Math. Phys. 25 (1972), 190--232.
• J. I. Hutchinson, On a remarkable class of entire functions, Trans. Amer. Math. Soc. 25 (1923), 325--332.
• O. Holtz, Hermite-Biehler, Routh-Hurwitz, and total positivity, Linear Algebra Appl. 372 (2003), 105--110.
• D. C. Kurtz, A sufficient condition for all the roots of a polynomial to be real, Amer. Math. Monthly 99 (1992), 259--263.
• N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963.
• D. G. Wagner, Total positivity of Hadamard products, J. Math. Anal. Appl. 163 (1992), 459--483.
• Symmetric Functions Catalog: https://www.symmetricfunctions.com/realRooted.htm, https://www.symmetricfunctions.com/realRootedInterlacing.htm, https://www.symmetricfunctions.com/polyaFrequency.htm, and https://www.symmetricfunctions.com/realRootedWords.htm.